Tutorials

• Alternating least squares for Canonical Polyadic (CP) Decomposition
  • Generate data
  • Basic call to the method, specifying the data tensor and its rank
  • Run again with a different initial guess, output the initial guess.
  • Increase the maximium number of iterations
  • Compare the two solutions
  • Rerun with same initial guess
  • Changing the output frequency
  • Suppress all output
  • Use HOSVD initial guess
  • Change the order of the dimensions in CP
  • Change the tolerance
  • Control sign ambiguity of factor matrices
  • Recommendations
• CP-APR: Alternating Poisson Regression for fitting CP to sparse count data
  • Set up a sample problem
  • Call CP-APR
• Generalized CP (GCP) Tensor Decomposition
  • Basic Usage
  • Specifying Missing or Incomplete Data Using the Mask Option
  • Solver options
  • Other Options
    • Specifying L-BFGS-B Parameters
    • Specifying SGD, Adagrad, and ADAM Parameters
      • Stochastic Gradient
      • Estimating the Function.
  • Example with Gaussian distribution
    • Run GCP-OPT
    • Compare to CP-ALS, which should usually be faster
    • Now let’s try is with the ADAM functionality
  • Create an example Rayleigh tensor model and data instance.
    • Run GCP-OPT
    • Now let’s try is with the scarce functionality - this leaves out all but 10% of the data!
  • Boolean tensor.
    • Convert ktensor to an observed tensor. Get the model values, which correspond to odds of observing a 1
      • Just for fun, let’s visualize the distribution of the probabilities in the model tensor.
    • Call GCP-OPT on the full tensor
    • Call GCP-OPT on a sptensor
  • Create and test a Poisson count tensor.
    • Loss function for Poisson negative log likelihood with identity link.
• Computing Tucker via the HOSVD
  • Higher-order Singular Value Decomposition (HOSVD) and Sequentially-truncased HOSVD (ST-HOSVD)
  • Simple example of usage
  • Generate a core with different accuracies for different shapes
  • Generate data tensor with core described above
  • Compute (full) HOSVD
  • Compute low-rank HOSVD approximation
  • Verbosity - Getting more or less information.
  • Example 1
  • Example 2
  • Specify the ranks
  • Specify the mode order
  • Generate bigger data tensor with core described above
  • ST-HOSVD compared to HOSVD
• Alternating least squares for Tucker model
  • Create a data tensor of shape (5, 4, 3)
  • Create an approximation with all ranks equal to 2
  • Create an approximation with specific ranks of [2, 2, 1]
  • Use a different ordering of the dimensions
  • Use the "nvecs" initialization method
  • Specify the initial guess manually
• Kruskal Tensors
  • Kruskal tensor format via ktensor
  • Specifying weights in a ktensor
  • Creating a one-dimensional ktensor
  • Constituent parts of a ktensor
  • Creating a ktensor from its constituent parts
  • Creating an empty ktensor
  • Use full or to_tensor to convert a ktensor to a tensor
  • Use double to convert a ktensor to a multidimensional array
  • Use tendiag or sptendiag to convert a ktensor to a ttensor
  • Use ndims and shape for the dimensions of a ktensor
  • Subscripted reference for a ktensor
  • Subscripted assignment for a ktensor
  • Using negative indexing for the last array index
  • Adding and subtracting ktensors
  • Basic operations with a ktensor
  • Use permute to reorder the modes of a ktensor
  • Use arrange to normalize the factors of a ktensor
  • Use fixsigns for sign indeterminacies in a ktensor
  • Use ktensor to store the ‘skinny’ SVD of a matrix
  • Displaying a ktensor
• Converting Sparse Tensors to Matrices and vice versa
  • Creating an sptenmat (sparse tensor as sparse matrix) object
  • Constituent parts of an sptenmat
  • Creating an sptenmat from its constituent parts
  • Creating an sptenmat with no nonzeros
• Creating an empty sptenmat
  • Use double to convert an sptenmat to a SciPy COO Matrix
  • Use full to convert an sptenmat to a tenmat
  • Use to_sptensor to convert an sptenmat to an sptensor.
• Sparse Tensors
  • Creating a sptensor
  • Specifying the accumulation method for the constructor
  • Creating a one-dimensional sptensor
  • Creating an all-zero sptensor
  • Constituent parts of a sptensor
  • Creating a sptensor from its constituent parts
  • Creating an empty sptensor
  • Use sptenrand to create a random sptensor
  • Use squeeze to remove singleton dimensions from a sptensor
  • Use squash to remove empty slices from a sptensor
  • Use full or to_tensor to convert a sptensor to a (dense) tensor
  • Use to_sptensor to convert a (dense) tensor to a sptensor
  • Use double to convert a sptensor to a (dense) multidimensional array
  • Use find to extract nonzeros from a tensor and then create a sptensor
  • Use ndims and shape to get the shape of a sptensor
  • Use nnz to get the number of nonzeroes of a sptensor
  • Subscripted reference for a sptensor
  • Subscripted assignment for a sptensor
  • Using negative indexing for the last array index
  • Use elemfun to manipulate the nonzeros of a sptensor
  • Basic operations (plus, minus, times, etc.) on a sptensor
    • Addition
    • Subtraction
    • Multiplication
    • Division
    • Equality
  • Use permute to reorder the modes of a sptensor
  • Displaying a sptensor
• Sum of Structured Tensors
  • Creating sumtensors
  • A Magnitude Example
  • Further examples of the sumtensor constructor
  • Ndims and shape for dimensions of a sumtensor
  • Use full to convert sumtensor to a dense tensor
  • Use double to convert to a numpy array
  • Matricized Khatri-Rao product of a sumtensor
  • Innerproducts of sumtensors
  • Norm compatibility interface
  • Use CP-ALS with sumtensor
  • Addition with sumtensors
  • Accessing sumtensor parts
• Converting a tensor to a 2D numpy array and vice versa
  • Creating a tenmat (tensor as 2D numpy array) object
  • Creating a tenmat by specifying the dimensions mapped to the rows
  • Creating a tenmat by specifying the dimensions mapped to the columns
  • Vectorize via tenmat
  • Alternative ordering for the columns for mode-\(n\) matricization
  • Constituent parts of a tenmat
  • Creating a tenmat from its constituent parts
  • Creating an empty tenmat
  • Use double to convert a tenmat to a 2D numpy array
  • Use to_tensor to convert a tenmat to a tensor
  • Use shape and tshape for the dimensions of a tenmat
  • Subscripted reference for a tenmat
  • Subscripted assignment for a tenmat
  • Using negative indexing for the last array index
  • Basic operations for tenmat
  • Multiplying two tenmats
  • Displaying a tenmat
• Tensors
  • Creating a tensor from an array
  • Creating a one-dimensional tensor
  • Specifying trailing singleton dimensions in a tensor
  • The constitutent parts of a tensor
  • Creating a tensor from its constituent parts
  • Creating an empty tensor
  • Use tenones to create a tensor of all ones
  • Use tenzeros to create a tensor of all zeros
  • Use tenrand to create a random tensor
  • Use squeeze to remove singleton dimensions from a tensor
  • Use double to convert a tensor to a (multidimensional) array
  • Use ndims and shape to get the shape of a tensor
  • Subscripted reference for a tensor
  • Subscripted assignment for a `tensor
  • Using negative indexing for the last array index
  • Use find for subscripts of nonzero elements of a tensor
  • Computing the Frobenius norm of a tensor
  • Using reshape to rearrange elements in a tensor
  • Basic operations (plus, minus, and, or, etc.) on a tensor
  • Using tt_tenfun for elementwise operations on one or more tensors
  • Use permute to reorder the modes of a tensor
  • Symmetrizing and checking for symmetry in a tensor
  • Displaying a tensor
• Tucker Tensors
  • Creating a ttensor with a tensor core
  • Creating a one-dimensional ttensor
  • Constituent parts of a ttensor
  • Creating a ttensor from its constituent parts
  • Creating an empty ttensor
  • Use full or to_tensor to convert a ttensor to a tensor
  • Use reconstruct to compute part of a full tensor
  • Use double to convert a ttensor to a (multidimensional) array
  • Use ndims and shape to get the shape of a ttensor
  • Subscripted reference for a ttensor
  • Subscripted assignment for a ttensor
  • Using negative indexing for the last array index
  • Basic operations (uplus, uminus, mtimes, etc.) on a ttensor
    • Addition
    • Subtraction
    • Multiplication
  • Use permute to reorder the modes of a ttensor
  • Displaying a ttensor
  • Partial Reconstruction of a Tucker Tensor
    • Benefits of Partial Reconstruction
    • Create a random tensor for the data.
    • Compute HOSVD
    • Full reconstruction
    • Partial reconstruction
    • Down-sampling
    • Compare visualizations